Purchase Solution

Prove that S = -k sum over r of P_r Log(P_r)

Not what you're looking for?

Ask Custom Question

Prove that k sum over r of P_r Log(P_r) for the canonical ensemble.

See attached file.

Attachments
Purchase this Solution

Solution Summary

A detailed explanation is given regarding a canonical ensemble.

Solution Preview

We need to express the entropy in terms of the partition function. This can be done as follows. It is fairly straightforward to express the pressure and internal energy of a system in terms of the partition function:

P = 1/beta d Log(Z)/dV (1)

E = -d Log(Z)/dbeta (2)

I'll give the proof below. Let's first see how we can derive an expression for the entropy using (1) an (2).
For a general differentiable function of two variables f(x,y) we have:

df = df/dx dx + df/dy dy

Where df/dx is the partial derivative of f w.r.t. x at constant y and df/dy is the partial derivative of f w.r.t. y at constant x.

So, if we take the function f to be Log(Z) and the variables x and y to be the volume V and temperature parameter beta, we can write:

d Log(Z) = d Log(Z)/dV dV + dLog(Z)/dbeta dbeta

Using (1) and (2) we can express the partial derivatives of Log(Z) in terms of the pressure and internal ...

Purchase this Solution


Free BrainMass Quizzes
Variables in Science Experiments

How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.

Classical Mechanics

This quiz is designed to test and improve your knowledge on Classical Mechanics.

Basic Physics

This quiz will test your knowledge about basic Physics.

Introduction to Nanotechnology/Nanomaterials

This quiz is for any area of science. Test yourself to see what knowledge of nanotechnology you have. This content will also make you familiar with basic concepts of nanotechnology.

The Moon

Test your knowledge of moon phases and movement.